"We will continue to play our part in ensuring that Europe remains strong and prosperous and able to lead in the world," Ms May wrote in the Irish Times a day after she launched Britain's withdrawal process.
She said Brexit was not an attempt "to do harm to the European Union or any of the remaining member states".
But she also reiterated her warning that failure to reach a trade deal would hamper security ties.
"It would be to the detriment of us all if unnecessary barriers to trade were erected," she said.
In Germany's Frankfurter Allgemeine Zeitung she underlined close cooperation on foreign policy matters such as the conflicts in Syria and Ukraine.
"We know how valuable this partnership and enduring friendship between our two countries are," she said, in comments translated from German.
"We appreciate hugely the friendship between our countries and our people," May said.
In France's Le Parisien newspaper May underlined the importance of "the closest possible security cooperation" after Britain leaves the EU.
She said that both sides "are faced with the same global threats of terrorism and extremism".
Ms May also wrote in Poland's Rzeczpospolita, Spain's El Pais and Sweden's Dagens Nyheter.
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